The ideas of the present paper have originated from the observation that all solutions of the linear homogeneous differential equation (DE) y″ (t) +y(t) = 0 satisfy the non-trivial linear homogeneous boundary conditions (BCs) y(0) + y(π) = 0, y′(0) + y′(π) = 0. Such a BC is referred to as a natural BC (NBC) with respect to the given DE, considered on the interval [0, π]. This observation suggests
... [Show full abstract] the following queries: (i) Will each second-order linear homogeneous DE possess a natural BC ? (ii) How many linearly independent natural BCs can a DE possess? The present paper answers these queries. It also establishes that any non-trivial homogeneous mixed BC, which is not a NBC with respect to the given linear homogeneous DE, determines uniquely (up to a constant multiplier), the solution of the DE. Two BCs are said to be compatible with respect to a given DE if both of them determine the same solution of the DE. Conditions for the compatibility of sets of two and three BCs with respect to a given DE have also been determined.